On the scaling of river network biogeochemical function

River networks play a fundamental biogeochemical role in the Earth system by transporting and processing materials from terrestrial to ocean ecosystems. The cumulative biogeochemical function of a watershed of area AW$$ {A}_W $$ can broadly be referred to as the total processing rate of material performed by its river network. An important recent research, conducted through network simulations, has revealed that the biogeochemical function of rivers can scale superlinearly with the area AW$$ {A}_W $$ under certain scenarios. This finding has significant implications for the role of river networks in regional and global biogeochemical cycles. Here, we demonstrate how such scaling can be derived analytically by combining the power law distribution of drainage area, the universal fractal signature of river networks and the scaling of channel hydraulic geometry, utilising the theory of finite-size scaling. The results enable the discrimination between linear and superlinear behaviours, as well as the calculation of the exact exponent based on parameters that define how the biogeochemical function and the river width change with river drainage area. Furthermore, we investigate the difference between the scaling of the biogeochemical function with the area of the watershed AW$$ {A}_W $$ and with the area of a region drained by multiple river networks, emphasising the implications for upscaling efforts.